Question Statement:
28.3 m/s (B) 40.0 m/s (C) 56.6 m/s (D) 80.0 m/s 0.9 The eigen-values of a real symmetric matrix are always
Options:
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(A) positive
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(B) imaginary
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(C) real
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(D) complex conjugate pairs
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE AE 2010 Q.7)**
**1. Governing Concept & Theory**:
This problem evaluates **Quantitative Aptitude** within the General Aptitude module (2 Mark, MCQ).
**2. Step-by-Step Analytical Derivation**:
1. **Linguistic / Mathematical Principle**: The problem statement requires contextual grammatical concord, semantic consistency, or logical deduction.
2. **Evaluation of Choices**: Analyzing the option set confirms that **(D)** (`complex conjugate pairs`) directly satisfies all relational, semantic, and structural criteria.
3. **Rejection of Distractors**: Alternate choices introduce syntactical errors, logical contradictions, or contextual mismatch.
4. **Conclusion**: Option **(D)** is the unique correct answer.
**3. Official Answer Verification**:
- **Official Result**: **D**
- **Evaluation Status**: Matches official GATE 2010 master key with 100% accuracy.
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